Strongly log-concave measures and lattices
Abstract
Let Λ⊆ ℝn be a full rank lattice with the property that det Γ≥ 1 for all sublattices Γ⊆ Λ*. We prove that there exists a symmetric random vector X ∈ ℝn with a smooth, strictly positive density f : ℝn → (0, ∞) satisfying that (1) f is α-strongly log-concave, where α= 2π/e, and (2) X + Λ is uniform on ℝn/Λ. Moreover, the existence of such a density characterizes this property up to a universal dilation. There are a few consequences of the above. We list some. First, for such lattices we can bound the covering radius μ(Λ, K) = maxx∈ℝnminλ∈Λ |x−λ|K ≤ √(2e/π) · 𝔼 |G|K of a convex body K ⊆ ℝn containing $0$ in its interior, where G ∼ N(0,Id). This implies an O(n √(log n)) bound for the flatness constant and an O(log3/2 n) bound for the subspace flatness constant. We also compare the entire ordered spectrum of the associated flat torus on Λ* with that of a Gaussian, yielding an alternative proof of the reverse Minkowski theorem.
Related
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- Tight Lower Bounds for Stochastic Nonconvex-Strongly-Concave Minimax Optimization
BibTeX
@misc{letwin2026strongly,
title = {{Strongly log-concave measures and lattices}},
author = {Brayden Letwin and Victor Reis and Thomas Rothvoss},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29238},
eprint = {2610.29238},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29238},
}
RIS
TY - UNPB
TI - Strongly log-concave measures and lattices
AU - Brayden Letwin
AU - Victor Reis
AU - Thomas Rothvoss
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.29238
UR - https://arxiv.org/abs/2610.29238
AB - Let Λ⊆ ℝ^n be a full rank lattice with the property that det Γ≥ 1 for all sublattices Γ⊆ Λ^*. We prove that there exists a symmetric random vector X ∈ ℝ^n with a smooth, strictly positive density f : ℝ^n → (0, ∞) satisfying that (1) f is α-strongly log-concave, where α= 2π/e, and (2) X + Λ is uniform on ℝ^n/Λ. Moreover, the existence of such a density characterizes this property up to a universal dilation. There are a few consequences of the above. We list some. First, for such lattices we can bound the covering radius μ(Λ, K) = max_x∈ℝ^nmin_λ∈Λ |x−λ|_K ≤ √(2e/π) · 𝔼 |G|_K of a convex body K ⊆ ℝ^n containing $0$ in its interior, where G ∼ N(0,Id). This implies an O(n √(log n)) bound for the flatness constant and an O(log^3/2 n) bound for the subspace flatness constant. We also compare the entire ordered spectrum of the associated flat torus on Λ^* with that of a Gaussian, yielding an alternative proof of the reverse Minkowski theorem.
ER -